Chemical Engineering Science 65 (2010) 6270–6281

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Fluid flow characteristics of vascularized channel networks

Kee-Hyeon Cho a,b, Moo-Hwan Kim a,n

a Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk 790-784, Republic of Koreab RIST (Research Institute of Industrial Science and Technology), Energy and Resources Research Department, Namgu, Pohang, Kyungbuk 790-600, Republic of Korea

a r t i c l e i n f o

Article history:

Received 31 May 2010

Received in revised form

9 September 2010

Accepted 10 September 2010Available online 17 September 2010

Keywords:

Mass transfer

Optimization

Laminar flow

Hydrodynamics

Constructal

Self-healing

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.09.020

esponding author. Tel.: +82 54 279 2165; fax

ail address: mhkim@postech.ac.kr (M.-H. Kim

a b s t r a c t

This paper reports the fluid flow characteristics of vascularized channel networks. To validate our

vascular designs by an analytical approach, three-dimensional numerical works were performed. The

numerical work covered the Reynolds number range of 2–1000, cooling channels volume fraction of

0.02, pressure drop range of 10–10 000 Pa, and six flow configurations: first, second, and third

constructal structures with optimized hydraulic diameters (D1 and D2) and non-optimized hydraulic

diameter (D) for each system size 10�10, 20�20 and 50�50, respectively. In these cases, the objective

was to compare global flow resistance and mass flow rate distribution of the analytical solutions with

those of numerical solutions subject to a fixed volume and a fixed pressure drop. This paper shows that

the fluid flow performance of the second constructs is superior to that of the first and third constructs

when the system size exceeds 20�20. The difference in flow resistance performance between the

optimized and non-optimized structures was found to increase and manifests itself clearly as

the system size increases. Results also reveal that flow uniformity become desirable with increasing the

system size, and that the third construct configurations have better flow uniformity than the other

architectures among the optimized and non-optimized channel configurations. The analytical results

are also compared with numerical data and good agreement between the numerical and analytical

results is found.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In a polymer electrolyte membrane fuel cell (PEMFC) system,during a high power density operation, up to 50% of the energyproduced is heat. Without proper thermal management, fuel cellperformance may not be maintained. A cell temperature that istoo high may lead to membrane dehydration, and a celltemperature that is too low may result in water condensationor flooding (Larminie and Dicks, 2000). Thus, it is important tominimize the pressure drop in the flow of coolant through thecooling plates, as the pumping power reduces the overallefficiency of the system and results in a more non-uniformtemperature, both of which are crucial to PEMFC performance(Lasbet et al., 2007).

Numerous investigators have studied thermal and flowcharacteristics in micro-channels and mini-channels. Gosselinand Bejan (2005) proposed the optimization of fluid networksbased on the minimization of the pumping power requirement.This work used the concept of constructal theory to develop adesign tool for minimizing the pumping power required to joinarbitrary points. Pence (2002) developed the fractal-like channelnetwork to yield a 60% lower pressure drop for the same total flow

ll rights reserved.

: +82 54 279 3199.

).

rate and a 30 1C lower wall temperature under identical pumpingpower conditions with identical total convective surface areas.Ryu et al. (2003) found that channel width and depth were morecrucial than others in relation to heat-sink performance amongvarious design variables. They also discovered that optimaldimensions and corresponding thermal resistance have a power-law dependence on the pumping power.

In general, the use of micro-channels may improve thermalperformance due to its compactness, whereas, it may also lead todisadvantages of increased pumping power and uneven tempera-ture distribution. Therefore, it is desirable to increase thetemperature uniformity of the wall while decreasing the pressuredrop to overcome the problems mentioned above. Note, however,that achieving better temperature uniformity and lower pumpingpower across the compact devices, such as the cooling plates ofPEMFC or micro-electromechanical systems (MEMS), simulta-neously is a difficult challenge.

In recent years, the optimization technology in multi-scaleheat sinks or cooling plates has attracted increasing interest.Several types of effective flow architectures are being proposed,because the design of flow architecture has a significant effecton the operating performance of engineering equipment. Animportant segment of this new literature is based on a generalprinciple: the constructal law (Bejan, 2000; Bejan and Lorente,2004, 2008), which was stated by Bejan in 1996 as follows: ’’For a

finite-size system to persist in time (to live), it must evolve in such a

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–6281 6271

way that it provides easier access to the imposed currents that flow

through it.’’

Constructal law focuses attention on the relationship betweenthe architecture of the flow system and its global performance.The constructal law and the use of tree architecture were firstproposed as a problem of pure heat conduction, and later wereextended to structures for convective fins, fluid flow, and heattransfer. The constructal theory is a mental mindset wherein thegeneration of flow structures that occur everywhere in nature(e.g. river basins, lungs, atmospheric circulation, and vasculartissues) can be reasoned based on the evolutionary principle ofthe increase in flow access over time. The constructal law isthe time direction of the ‘‘movie’’ of successive configurations(Bejan, 1997, 2000; Bejan and Lorente, 2004, 2008). When thedesigns evolve toward configurations that flow more easily, everydesign (an image) is replaced by an image that flows more easily.The succession of such images can be seen as the frames ofa movie. The movie tape runs toward images that flow moreeasily.

There is a growing volume of research on the development offlow architectures using the constructal law (e.g., Brod, 2003;Hernandez et al., 2003; Kraus, 2003; Bejan et al., 2004; Jones andGhassemi, 2004; Lundell et al., 2004; Senn and Poulikakos, 2004a,2004b; Tondeur and Luo, 2004; Muzychka, 2005; Pramanick andDas, 2005). One active direction in engineering is the vasculariza-tion of smart materials, so that they may offer new or improvedvolumetric functions: self-cooling, self-healing, and variabletransport properties (Wang et al., 2009). A body with embeddedtree-shaped flows that bathe the entire volume is a vascularizedbody. Recently, Kim et al. (2006), Lee et al. (2008), and Wang et al.(2006) illustrated how the method can be used to developvascular structures for self-healing materials, which are widelyencountered in natural systems, and can be inspired from thesesystems for design. Biomimetic design approaches are also beingused in the development of self-healing systems for polymercomposites (White et al., 2001; Rudraiah and Ng, 2004; Tooheyet al., 2007; Trask et al., 2007, 2008). An important new directionis the development of smart materials with self-healing function-ality. A continuous supply of healing agents throughout thevolume of the self-healing material calls for the vascularization ofthe entire volume, so that the entire volume of the structural

Fig. 1. Schematic diagram of the three-dimensional geometry of cooling plates with t

construct; and (c) third construct.

composite is protected against volumetric cracking. Otherwise,several cracks may form randomly, and at different sites, simul-taneously and repeatedly (Kim et al., 2006; Lee et al., 2008). Therecent progress on self-healing techniques points toward compo-site structures with embedded vasculatures of channels filledwith healing fluids. When cracks occur in the solid structure, thechannels are ruptured and the vasculature delivers the healingfluid to the crack sites. The vasculature is more effective when itdelivers the fluid faster and more uniformly regardless of the(random) position of the crack site. The present paper is about theflow characteristics of vasculatures of this kind, with minimalglobal flow resistance and minimal non-uniformity in how thefluid invades the volume.

Tree-shaped flow configurations offer maximum accessbetween one point (e.g. inlet or outlet) and an infinite numberof points (e.g. area or volume). Many superimposed tree flows areaccommodated by a grid of channels resembling the grid of citytraffic. For volumes that must be bathed uniformly by a singlestream flowing in and out, the recommended dendritic flowarchitecture consists of two trees matched canopy-to-canopy(Bejan, 2000) (Fig. 1(a)). Saber et al. (2009, 2010) discussed thegeometrical design of various channel networks based on a multi-scale approach and proposed rapid design of channel multi-scalenetworks with minimum flow maldistribution. New classes ofmulti-scale trees matched canopy to canopy are illustrated inFig. 1(b) and (c). The second construct in Fig. 1(b) has blocks ofparallel channels serving as canopies for two matched trees, andthe third construct in Fig. 1(c) has four blocks of parallel channels.This higher order construct has greater complexity.

In this paper, we consider three types of design on a square flatvolume consisting of 10�10 volume elements: a first-levelconstruct (Fig. 1(a)), a second-level construct (Fig. 1(b)), and athird-level construct (Fig. 1(c)). We analyze the hydrodynamicperformance of the optimized constructal channel configurations,evaluate their performance as vascular network systems for self-healing or self-cooling, and compare them with non-optimizedchannel configurations. We also validate their performance bycomparisons between the analytical and numerical results. Toachieve this, in addition to the structure with 10�10 elements,we compare the structures with 20�20 and 50�50 volumeelements.

ree-shaped cooling channel with 10�10 elements: (a) first construct; (b) second

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–62816272

2. Geometry

Consider the vascularized self-healing body consisting of asquare slab measuring X�Y and having the thickness W, where W

is the dimension of the solid body in the direction perpendicularto the plane X�Y, as illustrated in Fig. 1. The size of the squaredomain is measured in terms of N�N, where N is the number ofsmall square elements counted along one side (Cho et al., inpress). When different flow structures are compared, X and Y arefixed, but W (i.e., elemental length d) varies by increasing thesystem size, N2

¼10�10, 20�20, or 50�50. For example, Fig. 1represents a square slab with the system size, N2

¼10�10.The path to higher performance involves increasing the

freedom to morph the flow architecture (Bejan, 2000). Thisentails adding more length scales that can be varied; in this case,two diameter sizes instead of one. The ratio of diameters (D1/D2)is the additional degree of freedom. Optimized multiple scales D1

and D2 will be distributed non-uniformly through the availableflow volume, as illustrated in Fig. 1.

We built the first, second and third constructs with twodiameter sizes on a square domain with 10�10, 20�20, and50�50 elements from our earlier work (Cho et al., in press).According to constructal theory (Bejan, 2000; Bejan and Lorente,2004, 2008), the thinner channels (D1) should be placed in thecanopy and the thicker (D2) in the stem and main branches, asillustrated in Fig. 1. The shape of the channel cross-section isassumed to be fixed from one channel to the next. Note that theeffect of local pressure losses can be further reduced bysmoothing the transitions between the subsequent channels ofdifferent sizes. However, they are perpendicular at a channelintersection, while maintaining the original cross-section ofsubsequent channels for simplicity (Fig. 1).

We also define the corresponding non-optimized configurations,where channels have only one size, on a square domain composed of10�10, 20�20, and 50�50 elements. There is one channel sizeD, with channel volume Vc, fixed on the same basis as the results ofthe optimized configurations. The detailed geometrical dimensionsfor each configuration are summarized in Table 1.

The volume fraction occupied by all channels is held constant:

f¼total channel volume

total volume¼

Vc

Vð1Þ

where V is the total volume and Vc is the total channel flow volume.The void fraction f may be viewed as the average porosity of theentire structure, with the observation that, in these designs, thechannel volume is not distributed uniformly. Another geometricproperty of the flow structure is svelteness Sv, the ratio of theexternal length scale ((XY)1/2) divided by the internal length scale:

Sv¼external length scale

internal length scale¼ðXYÞ1=2

Vc1=3

ð2Þ

Table 1Geometric dimensions for the constructal configurations (X¼Y¼100 mm).

System

size

Complexity Vc

(mm3)

d

(mm)

D

(D1¼D2)

D2

(mm)

D1

(mm)

Sv

10�10 1st 2000 10 1.529 2.338 1.295 7.9

2nd 2000 10 1.476 1.979 1.045 7.9

3rd 2000 10 1.543 1.843 1.065 7.9

20�20 1st 1000 5 0.780 1.500 0.663 10.0

2nd 1000 5 0.764 1.282 0.537 10.0

3rd 1000 5 0.782 1.160 0.531 10.0

50�50 1st 400 2 0.316 0.842 0.274 13.6

2nd 400 2 0.313 0.737 0.228 13.6

3rd 400 2 0.316 0.656 0.222 13.6

Sv is a global property of the flow architecture, playing an importantrole in the evolution of the flow architecture towards the best, ornear-best, architecture in a fixed space (near the ‘‘equilibrium flowconfiguration’’ performance level (Bejan, 2000)).

3. Analytical model and method

3.1. Analytical model

We previously explored new vascular designs for the volu-metric bathing of smart structures with volumetric functionalities(self-healing, cooling). One stream bathes the volume, while thevasculature is configured as two trees matched canopy to canopy.Several architecture types are optimized: one channel size vs. twochannel sizes, increasing complexity (first, second, and thirdconstructs) and increasing size (up to 50�50 elemental volumes).A square domain composed of 10�10 elements for an analyticalapproach is illustrated in Cho et al. (in press). The centers of theelements are indicated with black circles. The only degree offreedom in the present study is the channel diameter ratio D1/D2.

The assumptions on which the analytical analysis is based arefully developed laminar flow, negligible pressure losses atjunctions or bends and fluid with constant properties.

The flow resistance for the Poiseuille flow through a straightduct with a polygonal cross-section can be written as

DP_m¼

nL

8V2

� �p2Po

Að3Þ

where n is the kinematic viscosity, p is the perimeter of the cross-section, Po is the Poiseuille constant (for example, Po¼16 forround cross-sections), and A is the area of the cross-section that isfixed, because the total duct volume V and the duct length L arefixed, namely A¼V/L (Bejan, 2000; Bejan and Lorente, 2008). Thegroup p2Po/A depends only on n, which is the number of sides ofthe polygonal cross-section, and accounts for how this lastgeometric degree of freedom influences global performance(Bejan and Lorente, 2008).

In summary, the pressure drop along a channel with length Li,diameter Di, and mass flow rate _mi is estimated as

DPi ¼ C_miLi

D4i

ð4Þ

where C is a constant factor, for example, C¼128n/p, if the ductcross-section is round and i indicates the channel rank, where DPi

denotes the corresponding pressure drop through a duct that caneither be a channel, a distributor or a collector portion (Bejan,2000; Kim et al., 2006; Bejan and Lorente, 2008; Lee et al., 2008).

3.2. Analytical method

The calculation of DP= _m consists of invoking Eq. (4) for allchannels, accounting for the mass continuity at every junction,and adding all the DPi’s along one flow path, from the inlet to theoutlet of the flow structure. The details of deriving the globaloptimized channel configurations are illustrated in an earlierwork (Cho et al., in press) and not repeated here.

The dimensionless global flow resistance is defined as

c¼DP

C _mf2d3 ð5Þ

where d is the elemental length and f is the porosity of a slab(Kim et al., 2006; Bejan and Lorente, 2008; Lee et al., 2008). Thedimensionless flow resistance parameter c, which can be relatedto the well-known Kleiber’s rule (Singer et al., 2004) because thespecific (mass-related) pressure drop across the optimized

Table 2Comparison of dimensionless pressure drop and dimensionless mass flow rate between analytical and numerical solutions for the first, second and third construct, with

two diameters.

System size Complexity Be WP (Pa) c¼ DPC _m f2d3� �

min(D1aD2) ~M

Analytical (a) Numerical (b) a�ba

�� ��(%) Analytical (c) Numerical (d) c�dc

�� ��(%)

10�10 1st �1.4�108�10 2.170 2.248 3.6 1.546�105 1.493�105 3.4

�1.3�1010�1,000 2.170 6.340 192.2 1.437�107 4.920�106 65.8

2nd �1.4�108�10 2.438 2.705 11.0 1.372�105 1.236�105 9.9

�1.3�1010�1,000 2.438 7.034 188.5 1.231�107 4.266�106 65.3

3rd �1.4�108�10 2.729 3.025 10.8 1.224�105 1.105�105 9.7

�1.3�1010�1,000 2.729 8.242 202.0 1.103�107 3.652�106 66.9

20�20 1st �4.1�108�30 1.884 2.088 10.8 2.685�105 2.423�105 9.8

�3.9�1010�3,000 1.884 4.248 125.5 2.556�107 1.134�107 55.6

2nd �4.1�109�30 1.848 2.071 12.1 2.731�105 2.437�105 10.8

�3.9�1011�3,000 1.848 4.371 136.5 2.518�107 1.064�107 57.7

3rd �4.1�109�30 2.184 2.298 5.2 2.309�105 2.194�105 5.0

�3.9�1011�3,000 2.184 5.099 133.5 2.115�107 9.060�106 57.2

50�50 1st �1.4�109�100 1.527 1.555 1.8 4.426�105 4.346�105 1.8

�1.3�1011�10,000 1.527 2.365 54.9 4.325�107 2.792�107 35.4

2nd �1.4�109�100 1.230 1.361 10.7 5.487�105 4.958�105 9.6

�1.3�1011�10,000 1.230 1.909 55.2 5.184�107 3.339�107 35.6

3rd �1.4�109�100 1.477 1.630 10.4 4.569�105 4.140�105 9.4

�1.3�1011�10,000 1.477 2.348 59.0 4.304�107 2.707�107 37.1

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–6281 6273

channel networks increases with decreasing the system size (N2),as illustrated in Table 2, represents the ratio of the overallpressure drop to the mass flow rate across the slab in the case ofthe same size and porosity. The c value calculated for eachassumed configuration has a single value. For example, the cvalue for Fig. 1(a) reaches the minimum value 2.17, see Table 2.Here the corresponding mass flow rate for each configuration canbe determined from Eq. (5), see Table 2.

The pressure loss DP is calculated by summing up all flowimperfections (Bejan and Lorente, 2008)

DP¼X

d

f4L

Dh

1

2rU2

� �d

þX

l

K1

2rU2

� �l

ð6Þ

where Dh and r are the hydraulic diameter and fluid density,respectively. The first summation refers to friction losses causedby wall friction due to the viscosity (both molecular andturbulent) of water in motion, and results from a momentumtransfer between the molecules of adjacent fluid layers moving atdifferent velocities. The geometrical factor f only depends on theshape of the channel. In this work, the roughness of the channelwalls is ignored.

The second summation in Eq. (6) refers to local losses, such aspressure drops caused by junctions, fittings, valves, inlets, outlets,enlargements, and contractions. Each local loss contributes to thetotal loss in proportion to KrU2/2, where K is the respective local losscoefficient. In general, local losses can be non-negligible in thefunctioning of complex tree-shaped networks. In other words, thefriction losses gain in relative importance to the local (junctions)losses, as the svelteness of the flow architecture increases.

4. Numerical model and method

4.1. Numerical model

The fluid flow performance of the constructal channel architec-tures are numerically simulated using a model for 3-dimensionalfluid flow for each configuration. To simplify the numericalsimulation, only the channel part of the vascular network body isincluded in the computational domain.

Self-healing agent or coolant is provided by an embeddedthree-dimensional channel network. It is pumped into the self-healing network with a specified pressure at the inlet, asillustrated in Figs. 1 and 2. This pressure must be high enoughto overcome the pressure losses when the self-healing networksystem is under operation. The porosity f is fixed at 0.02 for allconfigurations.

The numerical work covers the overall pressure drop rangeDP¼10–10 000 Pa, which corresponds to the mass flow rate range4.38�10�7–1.63�10�3 kg/s. The material properties of waterused in this study are: density r¼998.2 kg/m3 and dynamicviscosity m¼8.51�10�5 kg/m s.

To focus on the effect of the optimized and non-optimizedchannel configurations on the vascular network performance, thefollowing assumptions are made:

�

The fluid flow is steady-state, isothermal, and 3-dimensional. � The fluid is Newtonian and single phase, while the flow is laminar. � All the fluid properties are constant.

Based on the above assumptions, the governing equations formass and momentum were solved numerically in the fluiddomains of the cooling channels, as follows:

�

Mass conservation:

@u

@xþ@v

@yþ@w

@z¼ 0 ð7Þ

�

Momentum equation:

r u@u

@xþv

@u

@yþw

@u

@z

� �¼�

@p

@xþm @2u

@x2þ@2u

@y2þ@2u

@z2

!ð8Þ

r u@v

@xþv

@v

@yþw

@v

@z

� �¼�

@p

@yþm @2v

@x2þ@2v

@y2þ@2v

@z2

!ð9Þ

r u@w

@xþv

@w

@yþw

@w

@z

� �¼ -

@p

@zþm @2w

@x2þ@2w

@y2þ@2w

@z2

!ð10Þ

Velocity components in the x, y, and z directions in thecoordinate system were designated by u, v, and w, respectively.

Fig. 2. Schematic diagram showing a flow path for pressure distribution (10�10 elements): (a) first construct; (b) second construct; and (c) third construct.

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–62816274

The variables are defined in the Nomenclature. The work ofnumerically solving Eqs. (7)–(10) was based on a dimensionlessformulation using the variables:

~x ¼x

L, ~y ¼

y

L, ~z ¼

z

L, ~n ¼

n

Lð11Þ

~u ¼u

L

mDP

, ~v ¼v

L

mDP

, ~w ¼w

L

mDP

ð12Þ

~P ¼ðP�PoutÞ

DPð13Þ

where L is a reference length (L¼X¼Y) and ~u, ~v and ~w arethe velocity components in the ~x, ~y, ~z directions, while DP¼

Pin�Pout.The resulting dimensionless equations include:

�

Mass conservation:

@ ~u

@ ~xþ@ ~v

@ ~yþ@ ~w

@~z¼ 0 ð14Þ

�

Momentum equation:

~u@ ~u

@ ~xþ ~v

@ ~u

@ ~yþ ~w

@ ~u

@~z

� �¼�

1

Be

@ ~p

@ ~xþ

1

Be

@2 ~u

@ ~x2þ@2 ~u

@ ~y2þ@2 ~u

@~z2

!ð15Þ

~u@ ~v

@ ~xþ ~v

@ ~v

@ ~yþ ~w

@ ~v

@~z

� �¼�

1

Be

@ ~p

@ ~yþ

1

Be

@2 ~v

@ ~x2þ@2 ~v

@ ~y2þ@2 ~v

@~z2

!ð16Þ

~u@ ~w

@ ~xþ ~v

@ ~w

@ ~yþ ~w

@ ~w

@~z

� �¼�

1

Be

@ ~p

@~zþ

1

Be

@2 ~w

@ ~x2þ@2 ~w

@ ~y2þ@2 ~w

@~z2

!

ð17Þ

The dimensionless pressure drop number (Be) is defined by

Be¼DPL2

mn ð18Þ

where m and n are the fluid dynamic viscosity and fluid kinematicviscosity of the fluid, water, respectively. The dimensionlesspressure drop number (Be) that Bhattacharjee and Grosshandler(1988) and Petrescu (1994) termed the Bejan number is clearly ameasure of the relative magnitude of the heat transfer and fluidfriction irreversibility. The pressure drop number plays the samerole in forced convection that the Rayleigh number plays in

natural convection. The Be value is fixed, because DP is fixed inthis study.

No-slip boundary conditions are applied at the channelsurfaces. The channel surfaces are impermeable with no slip.The overall pressure drop across the self-healing networks isdefined as DP¼Pin�Pout, where Pin and Pout are the inlet andoutlet pressures, respectively. The lowest pressure (Pout¼0) ismaintained as constant and uniform over the outlet planes of allchannels. The resulting pressure drops across the coolingchannels were such that the Reynolds number based on the inletchannel diameter was in the 2–1000 range of Reynolds numbers.

4.2. Numerical method and grid independence

Computations were performed using a finite-volume package(Fluent Inc., 2006) with a pressure-based solver, node-basedgradient evaluation, Semi-Implicit Method for Pressure-LinkedEquations (SIMPLE) algorithm for pressure–velocity coupling andsecond order upwind scheme for momentum equations. In thegrid generation, tetra grids were adopted. The independence ofthe solution, with respect to the grid size, was checked byexamining the values of the mass flow rate and pressure dropbetween the inlet and outlet, and the skin friction coefficients foreach geometrical configuration. Grid independence tests werecarried out for the six configurations presented in the precedingsections. The number of cells varied from case to case; forexample, the smallest number was 5�106 for the optimized firstconstruct with 10�10 elements. Convergence is achieved whenthe residuals for the mass and momentum equation were smallerthan 10�6. In addition, we closely examined the differencebetween the incoming and outgoing mass flow. For all of thecases simulated in this study, the relative difference was keptwithin 0.01%.

5. Results and discussion

5.1. Hydrodynamic characteristics

Three-dimensional computational works were carried out topredict the fluid flow characteristics of the new vascular designsfor the volumetric bathing of the smart structures. To simplify theproblems, only the fluid flow performance of vascular channels,rather than the self-healing system or body, is discussed.

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–6281 6275

The analytical work illustrated that the pressure drop throughthe optimized vascular self-healing system was relatively lowerthan that through the non-optimized channels, given the identicalchannel volume and mass flow inlet. To verify the validity of ouranalytical methodology, the analytical results of the optimizedvascular designs were compared with the numerical results.

The total pressure drop versus the mass flow rate was found tobe one of the most important details in reference to the design of thecompact devices or micro-electromechanical systems (MEMS). Thepressure drop was defined as the difference of the area-weightedaverage static pressure of the inlet and the outlets as follows:

DP¼

RRinletP dARRinletdA

�

RRoutletP dARRoutletdA

ð19Þ

The local mean velocity at the inlet (U) was the mass-weightedaverage velocity and defined as

U ¼

RRinletrvin dARR

inletrdAð20Þ

where vn is the velocity vector through surface A at the inlet.The Reynolds number based on the inlet hydraulic diameter

(Din) and the local mean velocity at the inlet (U) is defined as

Re¼rUDin

m ð21Þ

The mass flow rate was also non-dimensionalized by writing inthe order of the magnitude sense that for Poiseuille flow in atube with length Y and hydraulic diameter D the pressure dropscales as

DP�_mnY

D4ð22Þ

The channel diameter scale is related to the porosity of thestructure:

f�D2Y

XYWð23Þ

where X�Y. From Eqs. (22) and (23) and the Be definition Eq. (18),the proper dimensionless group for the mass flow rate emerges as

~M ¼_mY

nrf2W2ð24Þ

Fig. 3. Comparison of the global flow resistances between the a

where n is the kinematic viscosity, r is the density, f is theporosity of cooling channel in a slab, W is the thickness ofvascularized body and Y is the length of the vascularized body.

Fig. 3 demonstrates the results of the comparison of thedimensionless global flow resistance c between the analyticaland numerical results. For the optimized designs, as illustrated inFig. 3(a), the global flow resistance c decreases slowly as thesystem size (N2) increases. Note, however, that the global flowresistance c increases steeply as the system size (N2) increases forthe non-optimized designs, as illustrated in Fig. 3(b). Largerarchitectures flowed more easily if they were configured accord-ing to the developed method. The dimensionless global flowresistance c for the optimized vascular designs also increased asthe overall pressure drop DP increased when the local losses wereconsidered in the numerical analysis. In particular, they increasedsharply for the system size, 10�10, because of the small Sv (7.6)and resulting vortices.

The analytical and numerical results were very close to eachother when the pressure drop number (Be) was relatively small.For example, the minimal relative difference between theanalytical and numerical results was approximately 3.6% in thevicinity of Beffi1.4�108 for the first construct with 10�10elements, as illustrated in Table 2. However, the third constructwith 10�10 elements led to a large discrepancy (202.0%) in thevicinity of Beffi1.3�1010. This phenomenon was also due to thehigh local pressure losses, when the pressure drop number (Be)increased. These results are summarized in Table 2, which detailsthe performance of the optimized architecture of the presentstudy on the square domains N�N that increased in size all theway to 50�50.

Table 3 illustrates the relative percentage difference betweenthe dimensionless mass flow rates of the optimized and non-optimized structures. In comparing the dimensionless mass flowrate of optimized and non-optimized constructs, we see thatthe design of the optimized constructs is superior to that of thenon-optimized constructs. For example, the dimensionless massflow rate of the optimized design is larger by 137.6% based on thenon-optimized construct in the vicinity of Beffi1.4�108. Themaximum relative percentage difference between the dimension-less mass flow rates of the optimized and non-optimizedstructures is also about 370.4% based on the non-optimizedconstruct for 20�20 elements. This trend is manifested moreclearly when the system size increases: the optimized secondconstruct designs for 50�50 elements provide greater flow

nalytical and numerical results: (a) D1aD2 and (b) D1¼D2.

Table 3Comparison of dimensionless mass flow rate between the optimized and non-optimized constructs from numerical solutions for the first, second and third construct, with

two diameters and one diameter.

System size Complexity Be ~M b�ab

��� ��� (%)

Optimized (D1aD2) (a) Non-optimized (D1¼D2) (b)

10�10 1st �1.4�108 1.493�105 6.284�104 137.6

�1.3�1010 4.920�106 2.200�106 123.6

2nd �1.4�108 1.236�105 5.812�104 112.7

�1.3�1010 4.266�106 2.195�106 94.4

3rd �1.4�108 1.105�105 6.740�104 63.9

�1.3�1010 3.652�106 2.416�106 51.2

20�20 1st �4.1�108 2.423�105 5.137�104 371.7

�3.9�1010 1.134�107 3.035�106 273.6

2nd �4.1�108 2.437�105 5.181�104 370.4

�3.9�1010 1.064�107 3.260�106 226.4

3rd �4.1�108 2.194�105 6.122�104 258.4

�3.9�1010 9.060�106 3.565�106 154.1

50�50 1st �1.4�109 4.346�105 3.223�104 1,248.4

�1.3�1011 2.792�107 2.685�106 939.9

2nd �1.4�109 4.958�105 3.346�104 1,381.8

�1.3�1011 3.339�107 2.919�106 1,043.9

3rd �1.4�109 4.140�105 3.876�104 968.1

�1.3�1011 2.707�107 3.260�106 730.4

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–62816276

access, i.e., larger by a factor of about 13 based on the non-optimized construct in the vicinity of Beffi1.4�109. These resultsare summarized in Table 3.

Fig. 4 illustrates how the local static pressure along thecenterline of the system (z¼0) varied under the identical flowrate conditions for the three system sizes (10�10, 20�20 and50�50), where the mass flow rate _m of 10�10 elements wasfixed at 5�10�4 kg/s, 20�20 elements at 3�10-4 kg/s and50�50 elements at 1�10-4 kg/s, respectively. For comparison,the flow path was defined in Fig. 2 by presenting a number alongone path from the lower-left to the upper-right corner of thediagram in a clockwise direction. As illustrated in Fig. 5, ‘‘0’’ is thestart point at the inlet, while the final point is ‘‘4’’ at the outlet forthe first and second constructs and ‘‘6’’ for the third constructs.

The non-dimensionalized length ~wð ~w ¼ w=LwÞwas also intro-duced to plot the curve. For example, the dimensional length Lw is0.19 m for the non-optimized first construct with 10�10elements. The constant pressure gradients observed in thereferenced channel (i.e. range ‘‘1–2’’ for the first, second, andthird constructs, range ‘‘2–3’’ for the second and third construct)exhibit convincing evidence of the fully developed flow, con-sidering that the left part of the curve corresponds to the inletposition. The non-linear pressure distribution in the inlet part(position ‘‘1’’) was caused by the entrance effect and significantlycontributes to the total pressure drop.

In comparing the pressure distribution of the optimized andnon-optimized constructs, we found that the design of theoptimized constructs was superior to that of the non-optimizedconstructs. For example, the pressure drop through the optimizedfirst constructs with 10�10 elements was approximately261.6 Pa lower than that through the non-optimized firstconstructs, whose value was around 131.4 Pa. Considering theseresults, a lower pressure drop across each channel for theoptimized constructs was anticipated to result in a better flowuniformity, while a higher pressure drop across each channel forthe non-optimized constructs was anticipated to yield a worseflow uniformity.

As observed in Fig. 4, there were clear pressure recoveries atthe first bifurcation position (1) for the non-optimized firstconstruct with 10�10 elements, as expected; 24.6 Pa afterdecreasing approximately 35.2 Pa from the inlet. The pressure

recovery at position (1) was caused by the sharp change andrecovery of the flow direction. A similar phenomenon occurred atthe outlet, due to the same reason, and the pressure gradient atthe outlet was the highest along the flow path, which resulted in arelatively large pressure drop. We also found that the difference ofpressure drops between the optimized and non-optimizedvascular designs increased significantly as the system size (N2)increased. Later, a similar phenomenon in hydrodynamic char-acteristics was also found.

The skin-friction coefficient Cf, based on the area-averagedwall shear stress, a non-dimensional parameter defined as theratio of the wall shear stress and the reference dynamic pressureis defined as

Cf ¼tw

12rvb

2ð25Þ

where tw is the area-averaged wall shear stress, and r and vb

are the reference density and velocity, respectively. Notice thatEq. (25) is in terms of bulk conditions. These conditions implyperforming a volume-weighted average, e.g.

vb ¼

RALiui dAi

Vcð26Þ

where i indicates the channel rank and ui denotes the correspond-ing fluid velocity through surface Ai and length Li of straightchannels.

Fig. 5 illustrates the skin friction coefficient versus theReynolds number (Re), based on the inlet hydraulic diameter forthe two system sizes (10�10 and 50�50). From Fig. 5, it can beobserved that the skin friction coefficient Cf decreased withincreasing Re number increases. In addition, the drag on the wallof the optimized constructs was much lower than that of the non-optimized constructs, except for the vascular designs with 10�10elements (Fig. 5). Furthermore, the skin friction coefficient Cf

increased gradually at the high Reynolds number, due to themuch higher mass flow rate, as compared to those of the non-optimized designs. Again, with increasing flow rates, the pressuredrop caused by the bend, or junction, becomes more relevant.

Figs. 6 and 7 illustrate how the non-dimensional mass flowrate ( ~M) varied with the imposed pressure difference (WP) for thetwo system sizes (10�10 and 50�50). As seen in Fig. 6, it

Fig. 4. Pressure distributions along one path (‘‘0–1–2–3–4–5’’ for the first and second constructs, and ‘‘0–1–2–3–4–5–6’’ for the third constructs) subject to a fixed mass

flow rate: (a) 10�10 elements at _m ¼ 5� 10�4 kg/s; (b) 20�20 elements at _m ¼ 3� 10�4 kg/s; and (c) 50�50 elements at _m ¼ 1� 10�4 kg/s.

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–6281 6277

becomes evident that the global mass flow rate ~M increases with thepressure drop number (Be) and that the mass flow rate variation isnon-linear with its gradient increasing with the pressure drop. Thenon-linear relationship of the pressure drop and mass flow rate canbe attributed to the junctions and the developing of the fluid flow ateach junction or bend. At junctions or bends, the flow is disturbedand secondary flow motions are initiated, which then decays as itpasses along the straight channel, such that the flow tends todevelop again before reaching the next junction. In other words,with increasing mass flow rates, the pressure drop caused by thebend or junction becomes more relevant, due to the higher massflow rates for the configurations with 10�10 elements. In fact,when Sv was smaller than 10; the effect of the local pressure losses(e.g. junctions, entrances) was non-negligible (Lorente and Bejan,2005). These Poiseuille-type losses result in the discrepancybetween the analytical and numerical results. On the contrary, thepressure drop caused by the bend, or junction becomes negligible forsystem size with 50�50 elements. Consequently, the results ofanalytical and numerical approaches are in relatively good agree-ment (see Fig. 7).

Note that the third constructal designs for the non-optimizedchannel configurations provide smaller global flow resistanceamong all system sizes (Figs. 6(b) and 7(b)). This means that thedesign with more branching levels is a good design. Figs. 6(b) and7(b) show that although with the additional pressure drop atbifurcations the overall pressure drop of the third constructs issmaller than that of the first and second constructal structures,the third constructs distribute the flow better than the first andsecond constructs. This is especially true when the sveltenessnumber is greater than 10 in the cases with 20�20 and 50�50elements.

5.2. Flow non-uniformity

Flow maldistribution was evaluated by using the flow non-uniformity ratio in the mass flow rate distribution in the multi-scale channels. The non-uniformity ratio r was defined as

r¼_m i

_mtð27Þ

Fig. 6. Comparison of the dimensionless mass flow rate versus the pressure drop number (Be) between the analytical and numerical results for 10�10 elements: (a) D1aD2 and

(b) D1¼D2.

Fig. 7. Comparison of the dimensionless mass flow rate versus the pressure drop number (Be) between the analytical and numerical results for 50�50 elements: (a) D1aD2 and

(b) D1¼D2.

Fig. 5. Variation of skin friction coefficients with Reynolds number: (a) 10�10 elements and (b) 50�50 elements.

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–62816278

where _m i is the mass flow rate, only along the thin channels (D1),and _mt was the total value of mass flow rates along the thinchannels (D1), with a given number of channels (No) on the squaredomains of Fig. 2. The variable i indicates the channel, wherechannel number (No) is 10, 9, and 4 for the first, second, and thirdconstructs with 10�10 elements, respectively. The flow rates _mi

belong to the channels that cross the transverse ascending fromthe left to the right on the square domains of each construct, as

illustrated in Fig. 2. The values of the mass flow rate in eachchannel are calculated from the integration of the flow velocityinside each tube as follows:

mi ¼

ZArui � dAi ð28Þ

where ui is the velocity vector through surface Ai of thin channels(D1) and _mi is the mass flow rate along the thin channels (D1). The

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–6281 6279

higher non-uniformity ratio r indicates a poor distribution of fluidflow along the thin channels (D1).

The results of the flow maldistribution in the thin channelswith increasing Be are illustrated in Figs. 8 and 9 for the twosystem sizes (10�10 and 50�50). In comparing the maldistribu-tion of optimized and non-optimized constructs, we see that thedesign of the optimized constructs is also superior to that of thenon-optimized constructs, as we discussed in the previoushydrodynamic characteristics.

As the pressure drop number (Be) increases, for both theoptimized and non-optimized configurations, the mass flow ratedistribution of the first, second and third constructs becomesgenerally undesirable, due to the minor losses when Be ( or DP)increases (see Figs. 8 and 9); thus, resulting in a more non-uniform flow in the thin channels. On the other hand, the thirdconstruct configurations have a better flow uniformity than theother architectures among the optimized and non-optimizedchannel configurations, whereas the first construct architecture isnot desirable, because the uniformity effect decreases rapidly asBe increases and the system size N2 increases, as illustrated inFigs. 8 and 9.

Similarly, the comparison between the analytical and numer-ical results indicates that there is a positive agreement for boththe optimized and non-optimized configurations when the

Fig. 8. Comparison of the flow uniformity between the analytical and numerical res

construct.

pressure drop number (Be) was relatively small. On the contrary,the difference between the analytical and numerical resultsstarted to increase when increasing the overall pressure drop(DP), because of the increase in local losses.

6. Conclusions

In the present study, we investigated the fluid flow perfor-mance of constructal architecture based on the mass flow rates( _m) for the given pressure drop (DP). Six flow configurations weredescribed in this paper, and include the first, second and thirdconstructal structures with optimized hydraulic diameters (D1

and D2) and non-optimized hydraulic diameters (D) for eachsystem size, 10�10, 20�20, and 50�50.

The validations were conducted by comparing the analyticalresults with the numerical results using a three-dimensional CFDapproach. Numerical works confirmed the validity of theoptimized designs by the analytical approach from our earlierwork. A good agreement between the analytical and numericalresults was demonstrated; more specifically, the relative devia-tions of the dimensionless mass flow rate ( ~M) were approximately1.8% for the first construct with 50�50 elements.

ults for 10�10 elements: (a) first construct; (b) second construct; and (c) third

Fig. 9. Comparison of the flow uniformity between the analytical and numerical results for 50�50 elements: (a) first construct; (b) second construct; and (c) third

construct.

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–62816280

The results imply that the fluid flow performance of the firstconstructal structure was superior to that of the second and thirdconstructal structures when N¼10. The best structure is thesecond constructal structure, when N¼20 and 50 in the optimizedconstructal configurations. We also found that the difference inflow resistance performance between the optimized and non-optimized structures increases and manifests itself clearly as N

increases. On the other hand, the best architecture in non-optimized configurations is the third construct, across all workingconditions. These results are attributed to the hydrodynamicadvantage documented in Figs. 4, 6 and 7. In general, the flowuniformity becomes desirable as N increases. The third constructconfigurations have better flow uniformity than the otherarchitectures among the optimized and non-optimized channelconfigurations, whereas the first construct architecture is notdesirable, because the uniformity effect decreases rapidly as Be

increases.The optimized constructal self-healing channels offer a much

better flow uniformity and a lower pressure drop than the non-optimized self-healing channels (conventional types with onediameter). Therefore, the new vascular designs can be used todesign the vascular channel networks of self-healing or self-cooling, microvascular lab-on-a-chip systems or microelectroniccooling applications resulting in uniform velocity distributions

between the channels. Further work is also required to validatethe flow characteristics of new vascular designs and to quantifythe effectiveness of using them on the self-healing system. Anexperimental study is in progress to confirm the thermalperformances of new vascular designs.

Nomenclature

A area, m2

Be pressure drop number, Eq. (18)d elemental length scale, mCf skin friction coefficient, Eq. (25)Dh hydraulic diameter, m, Eq. (6)Di channel diameter, mD1 hydraulic diameter of thin channels, mD2 hydraulic diameter of thick channels, mf friction factor, Eq. (6)K local loss coefficient, Eq. (6)Li length of ith channel, mLt total channel length, m, Table 1_m mass flow rate, kg/s_mi mass flow rate of the ith channel only along the thin

channels (D1), kg/s

K.-H. Cho, M.-H. Kim / Chemical Engineering Science 65 (2010) 6270–6281 6281

_mt total mass flow rate only along the thin channels (D1),kg/s

~M dimensionless mass flow rate, Eq. (24)N number of elemental sequences d�d in one directionNo number of thin channels, Fig. 2P local pressure, PaPin inlet pressure, PaPout outlet pressure, PaPo Poiseuille constant, Eq. (3)Re Reynolds number, Eq. (21)Sv svelteness number, Eq. (2)u, v, w velocity components, m s�1, Eqs. (7)–(10)U local mean velocity at inlet, m s�1, Eq. (21)V total volume, m3

vb volume-averaged bulk velocity, m s�1, Eq. (26)Vc total channel flow volume, m3

x, y, z Cartesian coordinatesX, Y, W dimensions of vascularized unit, m, Fig. 1

Greek symbols

DP pressure difference, Par fluid density, kg m�3

f porosity, equation (1)m fluid dynamic viscosity, kg/s mn fluid kinematic viscosity, m2 s�1

c non-dimensional global flow resistance, Eq. (5)tw area-averaged wall shear stress, N m�2, Eq. (25)

Subscripts

b bulki channel rankin inletmin minimumout outlet

Superscript

� dimensionless

Acknowledgment

The authors would like to gratefully thank Dr. A. Bejan for hisfruitful discussions and helpful suggestions.

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